Which of the following is an equivalent expression to ((2^(-3)*9^(3))^(-4))/(2^(9)*9^(-10)) with only positive exponents that has been generated by applying the properties of integer exponents?
A. (2^(-7)*9^(-9))/(2^(9)*9^(-10))
B. (2^(3))/(9^(2))
C. 2^(3)*9^(2)
D. (2^(12)*9^(-12))/(2^(9)*9^(-10))
To simplify the given expression and generate an equivalent expression with positive exponents, we can apply the property of negative exponents, which states that a^(-n) = 1/a^n.
((2^(-3)*9^(3))^(-4))/(2^(9)*9^(-10))
Using the property of negative exponents, we can rewrite 2^(-3) as 1/(2^3) and 9^(-10) as 1/(9^10):
((1/(2^3)*9^(3))^(-4))/(2^(9)*(1/(9^10)))
Simplifying the numerator:
(1/(8*9^(3))^(-4))/(2^(9)*(1/(9^10)))
Simplifying 9^3 as 9^3 = (3^2)^3 = 3^(2*3) = 3^6:
(1/(8*3^(6))^(-4))/(2^(9)*(1/(9^10)))
Simplifying 3^(6*4) as 3^(24):
(1/(8*3^(24)))/(2^(9)*(1/(9^10)))
Simplifying the denominator 9^10 as 3^(2*10) = 3^20:
(1/(8*3^(24)))/(2^(9)*(1/(3^20)))
Now, let's simplify the expression further:
(1/(8*3^(24)))/(2^(9)/(1/(3^20)))
Using the property of division of exponents, we can rewrite 2^(9) as 1/(2^(-9)):
(1/(8*3^(24)))/(1/(2^(-9))/(1/(3^20)))
Using the property of division of exponents again to simplify the expression:
(1/(8*3^(24)))/(1*(2^9)/(3^20))
Simplifying 2^9:
(1/(8*3^(24)))/((512)/(3^20))
Simplifying further:
(1/(8*3^(24)))*(3^20)/512
Simplifying 3^(24-20):
(1/(8))*(3^4)/512
Simplifying 3^4:
(1/(8))*(81)/512
Simplifying 81/512:
(1/8)*(81)/(512)
Simplifying 81/512:
9/512
Therefore, the equivalent expression with positive exponents generated by applying the properties of integer exponents is:
B. (2^(3))/(9^(2))
((2^(-3)*9^(3))^(-4))/(2^(9)*9^(-10))
((2^-3 * 3^6)^(-4))/(2^(9)/3^(20))
((2^-3 * 3^6)^(-4))/(2^(9)/3^(20))
2^12 * 3^-24 *2^-9 * 3^20
2^3 * 3^-4
2^3 * 9^-2
2^3 / 9^2 (agree :)